Related papers: Graphs with nonnegative curvature outside a finite…

We prove distance bounds for graphs possessing positive Bakry-\'Emery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits…

In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-\'Emery setting as well in the…

We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main…

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-\'Emery curvature. This leads to a geometric criterion for the existence of a perfect matching. Precisely, we show that any regular graph with non-negative…

We classify all cubic graphs with either non-negative Ollivier-Ricci curvature or non-negative Bakry-\'Emery curvature everywhere. We show in both curvature notions that the non-negatively curved graphs are the prism graphs and the M\"obius…

We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume…

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author.…

In the present paper, we apply Alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature in the sense of Higuchi. We obtain the metric classification of these…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly…

For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.

We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally…

In this paper, we establish Betti number estimates for graphs with non-negative Ollivier curvature, and for graphs with non-negative Bakry-\'Emery curvature, providing a discrete analogue of a classical result by Bochner for manifolds.…

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the…

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq…

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly…

We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

We give a discrete Bonnet Myers type theorem for the effective diameter assuming positive Ollivier curvature. We prove that this diameter bound is attained if and only if the graph is a cocktail party graph, a Johnson graph, a halved cube,…

We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a…

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if…